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Isoperimetric Inequalities on Surfaces of Constant Curvature

Published online by Cambridge University Press:  20 November 2018

Hsu-Tung Ku ,
Mei-Chin Ku  and
Xin-Min Zhang
Hsu-Tung Ku
Affiliation:
Department of Mathematics and Statistics University of Massachusetts Amherst, MA U.S.A.
Mei-Chin Ku
Affiliation:
Department of Mathematics and Statistics University of South Alabama Mobile, AL U.S.A.
Xin-Min Zhang
Affiliation:
Department of Mathematics and Statistics University of South Alabama Mobile, AL U.S.A.
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Abstract

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In this paper we introduce the concepts of hyperbolic and elliptic areas and prove uncountably many new geometric isoperimetric inequalities on the surfaces of constant curvature.

Keywords

51M1051M2552A4053C20Gaussian curvatureGauss-Bonnet theorempolygon, pseudo-polygonpseudoperimeterhyperbolic surfaceHeron’s formulaanalytic and geometric isoperimetric inequalities
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 49 , Issue 6 , 01 December 1997 , pp. 1162 - 1187
Copyright
Copyright © Canadian Mathematical Society 1997

References

1. Berger, M., Convexity. Amer. Math. Monthly (8) 97(1990), 650–678.Google Scholar
2. Bhala, G.S., Brahmagupta's quadrilateral. Math. Comp. Ed. (3) 20(1986), 191–196.Google Scholar
3. do Carmo, M.P., Differential Geometry of Curves and Surfaces. Prentice-Hall, Inc., Englewood Cliffs, N.J., 1976.Google Scholar
4. Coxeter, H.S.M., Regular Polytopes. 3rd. edn, Dover Publications, Inc., New York, 1973.Google Scholar
5. Florian, A., Extremum Problems for Convex Discs and Polyhedra. Handbook of Convex Geometry, Vol. A (Eds.: Gruber, P.M. and Wills, J.M.), North-Holland Press, Amsterdam, 1993. 178–222.Google Scholar
6. Kazarinoff, N.D., Geometric Inequalities. New Math. Library, Math. Assoc. America, 1961.Google Scholar
7. Ku, H.T., Ku, M.C. and Zhang, X.M., Analytic and Geometric Isoperimetric Inequalities. J. Geom. 53(1995), 100–121.Google Scholar
8. MacNab, D.S., Cyclic Polygons and Related Questions. Math. Gaz. 65(1981), 22–28.Google Scholar
9. Marsden, J.E. and Tromba, A.J., Vector Calculus. 2nd edn, W.H.Freeman and Company, 1981.Google Scholar
10. Meschkowski, H., Noneuclidean Geometry. Academic Press, New York, 1964.Google Scholar
11. Osserman, R., The isoperimetric inequality. Bull. Amer.Math. Soc. 84(1978), 1182–1238.Google Scholar
12. Robbins, D.P., Areas of Polygons Inscribed in a Circle. Discrete Comput. Geom. 12(1994), 223–236.Google Scholar
13. Schoen, R. and Yau, S.T., Differential Geometry. Beijing, China, 1980 Google Scholar
14. Tang, D., Discrete Wirtinger and isoperimetric type inequalities. Bull. Austral. Math. Soc. 43(1991), 467–474.Google Scholar
15. Zhang, X.M., Bonnesen-style inequalities and pseudo-perimeters for polygons. J. Geom., to appear.Google Scholar